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A scaling law for aeolian dunes on Mars,

Venus, Earth, and for subaqueous ripples

Philippe Claudin and Bruno Andreotti

Laboratoire de Physique et Mécanique des Milieux Hétérogènes

UMR CNRS 7636

ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France.

Abstract

The linear stability analysis of the equations governing the evolution of a flat sandbed submitted to a turbulent shear flow predicts that the wavelength λ at whichthe bed destabilises to form dunes should scale with the drag length Ldrag =

ρsρf

d.

This scaling law is tested using existing and new measurements performed in water(subaqueous ripples), in air (aeolian dunes and fresh snow dunes), in a high pressureCO2 wind tunnel reproducing conditions close to the Venus atmosphere and in thelow pressure CO2 martian atmosphere (martian dunes). A difficulty is to determinethe diameter of saltating grains on Mars. A first estimate comes from photographsof aeolian ripples taken by the rovers Opportunity and Spirit, showing grains whosediameters are smaller than on Earth dunes. In addition we calculate the effect ofcohesion and viscosity on the dynamic and static transport thresholds. It confirmsthat the small grains visualised by the rovers should be grains experiencing saltation.Finally, we show that, within error bars, the scaling of λ with Ldrag holds over almostfive decades. We conclude with a discussion on the time scales and velocities at whichthese bed instabilities develop and propagate on Mars.

Key words: dune, saltation, Mars, instabilityPACS: 45.70.Qj (Pattern formation), 47.20.Ma (Interfacial instability), 96.30.Gc(Mars)

Aeolian sand dunes form appealing and photogenic patterns whose shapeshave been classified as a function of wind regime and sand supply [1,2]. Thesedunes have martian ‘cousins’ with very similar features [3]. Grains can alsobe transported by water flows, and bed instabilities are commonly reported influmes, channels and rivers [4,5,6]. The same dynamical mechanisms controlthe formation of aeolian (both on Earth and Mars) dunes and subaqueousripples 1 from a flat sand bed. In a first section we describe the framework

1 Subaqueous ‘ripples’ are defined as patterns whose typical size is much smaller

Preprint submitted to EPSL 28 August 2018

http://arxiv.org/abs/cond-mat/0603656v2

in which this instability can be understood. A unique length scale is involvedin this description, namely the sand flux saturation length Lsat, which scaleson the drag length Ldrag =

ρsρfd, where d is the grain diameter and ρs/ρf the

grain to fluid density ratio. It governs the scaling of the wavelength λ at whichdunes or subaqueous ripples nucleate.

However, attempting to test the scaling laws governing aeolian sand transportand dunes characteristic size, one faces a major problem: the grain size doesnot vary much from place to place at the surface of Earth and the grain to airdensity ratio is almost constant. This narrow range of grain size composingthese dunes has a physical origin: large grains are too heavy to be transportedby the wind and very small ones – dust – remain suspended in air. As shown byHersen et al. [7,8], small-scale barchan dunes can be produced and controlledunder water, with a characteristic size divided by 800 – the water to air densityratio – with respect to aeolian sand dunes. The martian exploration and inparticular the discovery of aeolian patterns (ripples, mega-ripples and dunes)give the opportunity to get an additional point to check transport scaling laws,as the martian atmosphere is typically 60 to 80 times lighter than air. Finally,data from high pressure CO2 wind tunnel reproducing conditions close to theVenus atmosphere [9] as well as fresh snow barchan pictures [10,11] nicelycomplement and confirm the scaling relation.

A difficulty already risen in [12] is to determine the diameter of saltating grainson Mars. A first estimate is derived from the analysis of photographs of the soiltaken by the rovers Opportunity and Spirit next to ripple patterns, showinggrains whose diameters are smaller than on Earth dunes [13]. We derive inthe appendix the effect of cohesion and viscosity on the dynamic and statictransport thresholds. It confirms that the small grains visualised by the roversare experiencing saltation. Finally, we show that, within error bars, λ scaleswith Ldrag over almost five decades and discuss the corresponding time scalesand velocities at which these bed instabilities develop and propagate.

1 Dune instability

The instability results from the interaction between the sand bed profile, whichmodifies the fluid velocity field, and the flow that modifies in turn the sandbed as it transport grains. Let us consider a flat and symmetric bump asthat shown in figure 1. The fluid is accelerated on the upwind (stoss) sideand decelerated on the downwind side. This is schematized by the flow linesin this figure: they are closer to each other above the bump. This results inan increase of the shear stress τ applied by the flow on the stoss side of the

that the water height, whereas that of subaqueous ‘dunes’ is comparable to depth.

2

Fig. 1. Schematic of the instability mechanism showing the stream lines around abump, the fluid flowing from left to right. A bump grows when the point at whichthe sand flux is maximum is shifted upwind the crest. The shift of the maximumshear stress scales on the size of the bump. The spatial lag between the shear andflux maxima is the saturation length Lsat.

bump. Conversely, τ decreases on the lee side. Assuming that the maximumamount of sand that can be transported by a given flow – the saturated sandflux – is an increasing function of τ , erosion takes place on the stoss slopeas the flux increases, and sand is deposited on the lee of the bump. If thevelocity field was symmetric around the bump, the transition between erosionand deposition would be exactly at the crest, and this would lead to a purepropagation of the bump, without any change in amplitude (we call this the‘A’ effect, see below). In fact, due to the simultaneous effects of inertia anddissipation (viscous or turbulent), the velocity field is asymmetric (even on asymmetrical bump) and the position of the maximum shear stress is shiftedupwind the crest of the bump (the ‘B’ effect). In addition, the sand transportreaches its saturated value with a spatial lag, characterized by the saturationlength Lsat. The maximum of the sand flux q is thus shifted downwind thepoint at which τ is maximum by a typical distance of the order of Lsat. Thecriterion of instability is then geometrically related to the position at which theflux is maximum with respect to the top of the bump: an up-shifted positionleads to a deposition of grains before the crest, so that the bump grows.

The above qualitative arguments can be translated into a precise mathematicalframework, see e.g. [14,15]. For a small deformation of the bed profile h(t, x),the excess of stress induced by a non-flat profile can be written in Fourierspace as:

τ̂ = τ0(A+ iB)kĥ, (1)

where τ0 is the shear that would apply on a flat bed and k is the wave vectorassociated to the spatial coordinate x. A and B are dimensionless functionsof all parameters and of k in particular. The A part is in phase with the bedprofile, whereas the B one is out of phase, so that the modes of h and τ ofwavelength λ have a spatial phase difference of the order of λB/(2πA). Thisshift is typically of the order of 10% of the length of the bump as A and B aretypically of the same order of magnitude. Expressions for A and B have beenderived by Jackson and Hunt [16] in the case of a turbulent flow. As shown

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Fig. 2. Sand transport relaxation lengths in units of Ldrag as a function of therescaled wind velocity u∗/uth, as predicted by the theoretical model presented in[20]. The inset shows the same curves in linear scale. Starting from a vanishingflux, the sand transport first increases exponentially over a distance shown with ◦symbols. The dominant mechanism during this phase is the ejection of new grainswhen saltons collide the sand bed. The spatial corresponding growth rate divergesat the threshold and rapidly decreases with larger u∗. As the number of grainstransported increases, the wind is slowed down in the saltation curtain, until satu-ration. The distance over which the flux relaxes towards its saturated value is shownwith • symbols. Except very close to the threshold, the dominant mechanism is thenegative feedback of transport on the wind.

by Kroy et al. [17], A and B only weakly (logarithmically) depend on thewavelength so that they can be considered as constant for practical purpose.

If the shear stress is below the dynamical threshold τth, no transport is ob-served, hence the sand flux is null. Above this threshold, one observes on aflat bed a saturated flux Q which is a function of τ0. The fact that a windof a given strength can only transport a finite quantity of sand is due to thenegative feedback of the grains transported by the fluid on the flow velocityprofile – the moving grains slow down the fluid. This saturation process ofthe sand flux is still a matter of research [18,19,20]. We refer the interestedreader to [20,21] for a review discussion. For our present purpose, we need afirst order but quantitative description of the saturated flux. Following [20],Rasmussen et al. wind tunnel data [22] are well described by the relationship:

Q = 25τ0 − τth

ρs

√

d

gif τ0 > τth and Q = 0 otherwise. (2)

g is gravitational acceleration. The prefactor 25 has been adjusted to fit thedata and is reasonably independent of the grain size d. The equivalent of therelation (1) for the saturated flux on a modulated surface qsat can then be

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written as

q̂sat = Q(Ã + iB̃)kĥ, (3)

where, by the use of relation (2), the values of à and B̃ simply verify A/à =B/B̃ = 1− τth/τ0.

As for any approach to an equilibrium state, there exists a relaxation length– or equivalently a relaxation time – scale associated with the sand flux satu-ration. This was already mentioned by Bagnold who measured the spatial lagneeded by the flux to reach its saturated value on a flat sand patch [23]. Asaturation length Lsat in dune models has been first introduced by Sauermannet al. [24], where the dependence of Lsat on τ and in particular its divergenceas τ → τth has been put phenomenologically in the description. In fact, thesaturation length should a priori depend on the mode of transport – at leastwe are sure that it must exist in all the situations where there is an equilibriumtransport. As there can be different mechanisms responsible for a lag beforesaturation (delay to accelerate the grain to the fluid velocity, delay due to thesedimentation of a transported grain [25], delay due to the progressive ejectionof grains during collisions, delay for the negative feedback of the transport onthe wind, delay for electrostatic effects between the transported grains andthe soil, etc), the dynamics is dominated by the slowest mechanism so thatLsat is the largest among the different possible relaxation lengths. In the aeo-lian turbulent case, there exists a detailed theoretical analysis [20] providingthe dependence of the saturation length on the shear velocity (• in figure 2).The curve roughly presents two zones. Very close to the threshold, the slowestprocess is the ejection of grains during collision. It is thus natural to have adivergence of the saturation length at the threshold [24] as the replacement ca-pacity crosses 1 (see the calculation of the dynamical threshold in Appendix).From just above the threshold – say for u∗/uth & 1.05 – the saturation lengthgently increases (roughly linearly) with u∗. However, in the field, the meanwind strength varies from day to day, as seasons goes by. For practical pur-poses – e.g. photograph analysis – it is thus of fundamental importance todefine an effective saturation length, independent of the wind velocity. Fortu-nately, the velocity rarely exceeds ∼ 3 uth, so that, in the range of interest, thevelocity dependence on Lsat is subdominant. Our first important conclusion isthus that even though it should be remembered that Lsat slightly depends onu∗/uth in lab experiments, the dominant parameters are the grain size d andthe sand to fluid density ratio. The theoretical analysis [20] shows that Lsatscales on the drag length Ldrag =

ρsρf

d, which is the length needed for a grain

in saltation to be accelerated to the fluid velocity. It is worth noting that thisinertial effect is however not the mechanism limiting the saturation process(see above). Using the results of a field experiment performed in the AtlanticSahara of Morocco [15], the prefactor between Lsat and Ldrag can be computedand gives:

Lsat ≃ 4.4ρsρf

d. (4)

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Fig. 3. Measurement of the typical grain size from the auto-correlation functionC(δ) of a photograph of the granular bed. a Sample of aeolian sand from themega-barchan of Sidi-Aghfinir (Atlantic Sahara, Morocco), of typical diameterd ≃ 165 µm. Top inset: Probability distribution function of the grain diameterd, weighted in mass. The narrow distribution around the maximum of probability ischaracteristic of the aeolian sieving process. Bottom inset: 200× 200 pix2 zoom ona typical photograph on which the computation of the autocorrelation function hasbeen performed. The image resolution is here 5 pix/d. b Sample of sand from Mars(at rover Spirit’s landing site: 16.6◦ S, 184.5◦ W). Top inset: photograph showingthe presence of milimetric grains as well as much smaller ones which are expected tobe the saltons. Bottom inset: same as a, but the typical image resolution is 3 pix/d.

This result is also consistent with Bagnold’s data [23]. Note that the saturationlength has never been measured directly in other situations (neither underwater nor in high/low pressure wind tunnels).

The linear stability analysis of the coupled differential equations of this frame-work has been performed in [14]. In particular, the wavenumber correspondingto the maximum growth rate is given by

kmax Lsat = X−1/3 −

X1/3

3with X =

3B̃

Ã

[

−3 +

√

3(

1 + (Ã/B̃)2)

]

. (5)

For typical values of à and B̃ determined in the barchan dune context [15],we have λmax = 2π/kmax ∼ 12Lsat. Note that we have A/B = Ã/B̃, so thatkmax Lsat is independent of τ0 and τth. Using measurements in water (subaque-ous ripples), in air (aeolian dunes) and in the CO2 atmosphere of Mars and ofthe Venus wind tunnel, we now investigate the scaling relation between λmaxand Ldrag. To do so, we need to first solve the controversy concerning the sizeof saltating grains on Mars.

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2 Size of saltating grains on Mars

The determination of the typical size of the grains participating to saltation onmartian dunes is a challenging issue. First, the dunes seem to evolve very slowlyor may even have become completely static. Second, no sample of the mattercomposing the bulk of the dunes is available. Third, we do know from theobservation of dunes on Earth that they can be covered by larger grains that donot participate to the transport in saltation. With the two rovers Opportunityand Spirit, we now have direct visualisations of the soil [13], and in particularof clear aeolian structures like ripples 2 or nabkhas 3 . Unfortunately, thesestructures did not lie on the surface of a dune. We thus analyze the availablephotographs, assuming that, like on Earth, the size of the grains participatingin saltation does not vary much from place to place.

2.1 Direct measurement of grain sizes

The photographs freely accessible online are not of sufficiently good resolu-tion to determine the shape and the size of each grain composing the surface.Besides, one has to be careful when analyzing such pictures as part of what isvisible at the surface corresponds to grains just below it and partly hidden bytheir neighbours. We have thus specifically developed a method to determinethe average grain size in zones where the grains are reasonably monodispersed,when the resolution is typically larger than 3 pixels per grain diameter. Thismeasure can be deduced from the computation of the auto-correlation func-tion C(δ) of the picture, which decreases typically over one grain size. Moreprecisely, we proceed with the following procedure:• The zones covered by anything but sand (e.g. gravels or isolated largergrains) are localized and excluded from the analysis.• Because in natural conditions the light is generally inhomogeneous, we per-form a local smoothing of the picture with a gaussian kernel of radius ≃ 10 d.The resulting picture is subtracted from the initial one.• We compute the local standard deviation of this image difference with thesame gaussian kernel and produce, after normalisation, a third picture I ofnull local average and of standard deviation unity.• The auto-correlation function C(δ) is computed on this resulting picture,

2 Contrarily to dunes, aeolian sand ripples form by a screening instability relatedto geometrical effects [23,26].3 As there is a reduction of pressure in the lee of any obstacle, sand accumulates inthe form of streaks aligned in the direction opposite to the wind (shadow dunes).These structures are called nabkhas [27].

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averaging over all directions:

C(δ) =

∑

k,l/k2+l2=δ2∑

i,j Ii,jIi+k,j+l∑

k,l/k2+l2=δ2∑

i,j 1(6)

Figure 3a shows the curve C(δ) obtained from a series of photographs of aeo-lian sand sampled on a terrestrial dune. For all the resolutions used (between1 and 5 pix/d), the data collapse on a single curve, which is thus characteristicof the sand sample. The top inset shows the distribution of size, weighted inmass, in log-log representation. It presents a narrow peak around the d50 value.The autocorrelation function decreases from 1 to 0 over a size of the order ofd50 (Lorenzian fit). This is basically due to the fact that the color or the graylevel at two points are only correlated if they are inside the same grain. It isthus reasonable to assume that the autocorrelation curve is a function of δ/d50only. As a matter of fact, photographs of two samples of comparable polydis-persity are similar once rescaled by the mean grain diameter. We will thus usethe curve obtained with aeolian sand sampled on Earth as a reference to de-termine the size of Martian grains. Figure 3b shows the autocorrelation curveobtained from the colour image taken by the rover Spirit at its landing site.C(δ) decreases faster than the reference curve but by tuning the value of themartian d50, one can superimpose the Mars data with the solid line fairly well.From this picture as well as a series of gray level photographs taken by therover Opportunity, we have estimated the diameter of the grains composingthese aeolian formations to 87± 25 µm.

The first measurement of martian grain sizes dates back to the Viking missionin the 70s. On the basis of thermal diffusion coefficient measurements, Edgettand Christensen have estimated the grain diameter to be around 500 µm andat least larger than those composing dunes on Earth [12]. This size correspondsto larger grains than saltons such as those shown in the top inset of figure 3b. Inagreement with our findings, wind tunnel experiments performed in ‘martianconditions’ [28] have shown that grains around 100 µm are the easiest todislodge from the bed. We shall come back to this point later on.

Several theoretical investigations of the size of aeolian martian grains havebeen conducted, starting with the work of Sagan and Bagnold [29]. In thatpaper the authors argue that, as Mars is a very arid planet, the cohesion forcesdue to humidity can be neglected and proposed a cohesion-free computationwhich predicts that very small grains (typically of one micron) may be put intosaltation. Miller and Komar [30] also followed this cohesion-free approach andproposed static threshold curves with no turnup on small particle size. It wassoon realized that cohesion forces can occur for reasons other than humidity –namely van der Waals forces – and several authors proposed (static) transportthreshold curves with a peaked minimum around 100 µm [31,32,33,34,35].However in these papers, cohesion is treated in an empirical fashion, with the

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Fig. 4. Diagram showing the mode of transport on Mars as a function of the graindiameter d and of the turbulent shear velocity uth (left) or of the wind speed Uth at2 m above the soil (right). Below the dynamical threshold (dashed line), no grainmotion is observed. A grain at rest on the surface of the bed starts moving, draggedby the wind, when the velocity is above the static threshold (solid line). Between thedynamical and static thresholds, there is a zone of hysteresis where transport cansustain due to collision induced ejections. The progressive transition from saltationto suspension as wind fluctuations become more and more important is indicatedby the gradient from white to gray color. See appendix for more details on thederivation of this graph.

assumption that van der Waals forces lead to an attractive force proportionalto the grain diameter with a prefactor independent of d.

We have recomputed the Martian transport diagram (figure 4) using a newderivation of the transport thresholds. It takes into account the hysteresisbetween the static and dynamic thresholds, the effect of viscosity and – ina more rigorous way – the effect of cohesion. As this derivation, althoughconsequent and original, is not the central purpose of the present paper whichis devoted to the test of the dune scaling law, we have developed and discussedit in appendix.

We wish to solely discuss here the figure 4, which is useful to prove that the87 µm sized grains can be transported in saltation. The first striking featureof the Martian transport diagram is the huge hysteresis between the dynamicand static thresholds. Compared to aeolian transport on Earth (figure A.2in the Appendix) for which the static threshold is typically 50% above thedynamic threshold, they are separated on Mars by a factor larger than 3.Quantitatively, if the static threshold is very high compared to the typicalwind velocities on Mars (∼ 150 km/h at 2 m above the soil), the dynamicalthreshold is only of the order of ∼ 45 km/h at 2 m. From images by theMars Orbiter Camera (MOC) of reversing dust streaks, Jerolmack et al. [36]have estimated that modern surface winds can reach velocities as large as

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Earth Ê Mars Ä water ê snow A ‘Venus’ Ã

g (m/s2) 9.8 3.7 9.8 9.8 9.8

λ 20 m 600 m 2 cm 15–25 m 10–20 cm

d (µm) 165 – 185 87 150 1500 110

ρf (kg/m3) 1.2 1.5 – 2.2 10−2 103 1.2 61

ρs (kg/m3) 2650 3000 2650 360 2650

ν (m2/s) 1.5 10−5 6.3 10−4 10−6 1.5 10−5 2.5 10−7

Table 1Comparison of different quantities (gravity g, initial wavelength of bed instabilityλ, diameter of saltons d as well as fluid and sediment densities ρf and ρs) in theair (Earth), in the martian and Venus wind tunnel CO2 atmospheres and in water.As the temperature at the surface of Mars can vary by an amplitude of typically100 K between warm days and cold nights, the density of the atmosphere displayssome variation range.

150 km/h. Looking at figure 4, it can be seen that at such velocities, most of thegrain below 100 µm can be suspended and that even millimetric grains can beentrained into saltation. Even in less stormy conditions, sand transport shouldnot be as unfrequent as one could expect, even though the large amplitude ofthe hysteresis implies an intermittency of sand transport. This suggests thatMartian dunes are definitively active.

3 A dune wavelength scaling law

Keeping in mind that the wavelength λ that spontaneously appears when aflat sand bed is destabilized by a turbulent flow scales on the flux saturationlength, the aim of the paper is to plot λ against Ldrag in the different situationsmentioned in the introduction: aqueous ripples, waves on aeolian dunes, bothon Earth and Mars, fresh snow dunes and microdunes in the Venus windtunnel – numerical values of the parameters corresponding to these differentsituations are summarized in table 1.

Our reference data point in figure 5 (aeolian dunes on Earth) has been ob-tained after an extensive work on barchan dunes in the Atlantic Sahara ofMorocco. In a recent paper [15], we have shown that perturbations such aswind changes generate waves at the surface of dunes by the linear instabil-ity described above. Direct measurements of the wavelength are reported inFigure 6a in an histogram. They give λ ∼ 20 m for waves on the flanks ofmedium sized dunes (solid line) and ∼ 28 m on the windward side of a mega-barchan (dashed line), where some pattern coarsening occurs. Lsat ∼ 1.7 m

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Fig. 5. Average wavelength as a function of the drag length. The two black squares(resp. diamonds) labeled ‘Earth’ (Ê) (resp. ‘Mars’ (Ä)) come from the two differenthistograms of figure 6a (resp. 6b). The three white circles are the under water (ê)Coleman and Melville’s data [5,6], whereas the black circle is that from Hersen etal.’s experiments [7,8]. The white triangle has been computed from the Venus (Ã)wind tunnel study [9]. Snow (A) dune photos (see figure 8) have been calibratedand complemented with data from [10,11,37], and give the white squares.

Fig. 6. Histograms of the wavelength measured in the Atlantic Sahara (Morocco)and in Mars southern hemisphere (mainly, but not only, in the region 320–350◦W,45–55◦S). (a) Earth (Ê). Solid line: wavelengths systematically measured on barchandunes located in a 20 km × 8 km zone; Dash line: wavelengths measured on thewindward side of a mega-barchan whose surface is permanently corrugated andwhere some coarsening occurs. (b) Mars (Ä). Solid line: wavelengths measured ondunes in several craters (e.g. Rabe, Russell, Kaiser, Proctor, Hellespontus); Dashline: histogram restricted to Kaiser crater (341◦W, 47◦S). Averaged values of λ are20 m (solid line on panel a), 28 m (dash line on panel a), 510 m (solid line on panelb) and 606 m (dash line on panel b).

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Fig. 7. Comparison of dune morphologies on Earth and on Mars. a Mega-barchanin Atlantic Sahara, Morroco. b ‘Kaiser’ crater on Mars (341◦W,47◦S). c Closer viewof the Kaiser crater dunes. As on Earth, small barchans are visible on the side ofthe field.

Fig. 8. Fresh snow aeolian dunes on ice. (a) Transverse dunes on the iced balticsea (credits Bertil H̊akansson, Swedish Meteorological and Hydrographical Insti-tute/Baltic Air-Sea-Ice Study). The wavelength is around 15 m [37]. (b) Snowbarchan field in Antartica (credits Stephen Price, Department of Earth and SpaceSciences, University of Washington). The shadow of the Twin Otters, which has hasa wing span of 19.8 m and a length of 15.7 m, gives the scale. In both pictures, theperspective has been corrected to produce a pseudo-aerial view.

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was measured independently in the same dune field, and these aeolian grainsof 180 µm lead to a drag length of 40 cm.

The study of small scale barchans under water [7,8] provides the second datapoint: measurements give λ ∼ 2 cm with glass beads of size d = 150 µm, asize which leads to a drag length of 400 µm. This point is the only one in theliterature for which it is specified that: (i) the wavelength has been measuredduring the linear stage of the instability; (ii) the sand bed destabilizes ina homogeneous way and not starting at the entrance of the set-up, due tostrong disturbances there; (iii) the height of the tank is much larger than thewavelength. We have added three other underwater points in figure 5, fromColeman and Melville’s data [5,6].The slight discrepancy with the straightline (these points are all above the line) can be explained by the three pointsrisen above: it may be induced from [5] that (i) the initial stage is not clearlyresolved; (ii) the waves seem to nucleate on defects as if the transition wassub-critical; (iii) the flume is only 0.28 m deep, a height which is comparableto the observed wavelengths.

Recent photos of martian dunes [3] such as that in figure 7 lead to an estimateof the value of λ ∼ 600 m on Mars. We focused on the dunes found in cratersof the southern hemisphere. For comparison, we have measured wavelengthson dunes in several craters (e.g. Rabe, Russell, Kaiser, Proctor, Hellespontus)and also produced a histogram restricted to Kaiser crater, see figure 6b. Asfor the drag length, we used the value for the grain diameter estimated in theprevious section. The density of martian grains is similar to or slightly higher[36] than that of terrestrial grains. The value of the density of the martianatmosphere however varies within some range as the temperature on Marssurface can change by an amplitude of typically 100 K between warm daysand cold nights. In the end, it gives a value of Ldrag between 13 and 17 m.

The fourth data point is obtained from Greeeley et al.’s experiment in a highpressure CO2 wind tunnel [9], which gives for Ldrag a value we expect onVenus. The observed wavelength first decreases from 18 cm to 8 cm with thewind speed from u∗ = uth to u∗ = 1.6 uth and then increases up to 27 cm atu∗ = 2.1 uth. Above this value, the flat sand bed was again found to be stable.Although some of these features are reminiscent of the Lsat curve discussedabove (figure 2), this series of experiments is also questionable: nothing isreported about the nature of the destabilization (homogeneous appearance ofthe pattern or not) nor on the maturity of the pattern when the wavelengthwas measured (coarsening?). Remarkably, the crude averaged of the provideddata lies on the master curve, even though the smallest wavelength is less thanhalf the predicted value.

As for the Venus wind tunnel, the fifth data points are not very precise andcorrespond to fresh snow dunes formed on Antartic sea ice and on Baltic sea

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ice. The typical wavelength λ of transverse dunes (for instance in figure 8(a)ranges from 15 m to 25 m [37]. Whenever barchan dunes form, they are typi-cally 5 m to 10 m long and are separated by ∼ 20 m (see figure 8(b). This isvery similar with aeolian ones, although snow barchan dunes look more elon-gated. The determination of Ldrag is more problematic as the snow densityrapidly increases once fallen. Snow dunes are rather rare and probably formonly at the surface of ice by strong wind, when the snow can remain fresh andnot very cohesive. To get the numbers, we have used the measurements per-formed close to the Antartic dunes by Massom et al. [10,11]. Although one mayfind somehow disappointing that on the graph of figure 5 these data points arelocated very close to the aeolian sand dunes, they are particularly interestingas they show that one can keep the same λ by changing simultaneously ρs andd in a compensating way.

Globally, we obtain a consistent scaling law λ ≃ 53Ldrag that covers almostfive decades (figure 5). We wish to emphasize again that we do not claimto capture all the dependences on this plot, but that Ldrag is the dominantscaling factor. As shown in figure 2, we expect subdominant dependences onthe wind speed, on finite size effects, etc... Note that this analysis explains forexample why martian dunes whose sizes are of the order of a kilometer arenot ‘complex’ or ‘compound’ as their equivalents on Earth [38]. Following ourscaling relation, they correspond to the small terrestrial dunes and the wholedune field on the floor of a crater should rather be considered as a complexmartian dune. A ‘similarity law’ for the size of (developed) dunes with Lsatwas schematically drawn by Kroy et al. [39], supported by the existence ofcentimetric barchans under water [7]. However, this similarity was announcedto fail with Martian dunes. In fact, the scaling of large dunes (i.e. whose sizesare much larger than the elementary length λ) with Ldrag (or Lsat) is far fromobvious as the size selection of barchans involves secondary instabilities relatedto collisions and fluctuations of wind direction [15,40,41].

Finally, we would like to end this section with the prediction of the wavelengthat which a flat sand bed should destabilize on Titan where (longitudinal orseif) dunes have been recently discovered [42]. The atmosphere there is ap-proximately four times denser than on Earth, and the grains are believed tobe made of water ice [43]. Computing the threshold curves (not shown), oneobserves that the dynamical and static thresholds are almost identical andthat the minimum of threshold shear stress is reached for 160 µm. Using thissize for the grain diameter a density ratio of the order of 200, we find a cen-timetric drag length. Following the scaling law of figure 5, this would lead toa dune wavelength between 1 and 2 meters. Note that, unfortunately, this iswell below the resolution of Cassini radar.

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4 Discussion: time scales and velocity for the martian dunes

Now that we have this unique length scale at hand for dunes, it is interesting toaddress the question of the corresponding growth time scales and propagationvelocities. The linear stability analysis [14,15] tells that the growth rate σand the propagation velocity c of bedforms are related to the wavenumberk = 2π/λ as

σ(k) = Qk2B̃ − ÃkLsat1 + (kLsat)2

and c(k) = QkÃ+ B̃kLsat1 + (kLsat)2

, (7)

where Q is the saturated sand flux – assumed constant – over a flat bed.In order to make these relations quantitative for the bedforms on Earth andMars, we need an effective time averaged flux Q̄. Recall from equation (2)that Q can be related to the shear velocity u∗. For simplicity, we suppose thatthere is sand transport (u∗ > uth) a fraction η of the time and that the shearstress is then constant and equal to (1+α)ρfu

2th. The time averaged sand flux

can thus be effectively expressed as

Q̄ = 25αη

√

d

g

ρfρs

u2th. (8)

The values of the coefficients à and B̃ which come into equation (7) dependon the excess of shear above the threshold as Ã/A = B̃/B = (1 + α)/α. Thetimescale over which an instability develops is that of the most unstable mode.This means that σ and c should be evaluated at k = kmax (see equation (5))and thus scale as:

σ∝Q̄

L2sat∝ (1 + α) η

(

ρfρs

)3u2th

g1/2d3/2, (9)

c∝Q̄

Lsat∝ (1 + α) η

(

ρfρs

)2u2th

g1/2d1/2. (10)

Using meteorological data and measurements of the average dune velocities,we estimated thatQÊ is between 60 to 90 m

2/year [15] and ηÊ between 65% and85%. This gives an effective value αÊ between 1.5 and 2. Calculating explicitlythe prefactors, we get a growth time σ−1Ê ∼ 2 weeks and cÊ ∼ 200 m/year,values which are consistent with direct observation. Note that time scales forindividual fully developed dunes depend on their size, but are also proportionalto 1/Q̄ [40].

In order to compare these terrestrial values to those on Mars, we need toestimate the different factors which come into the expressions (9) and (10).Recall that, on Earth, for grains of 165 µm, we have uÊth ∼ 0.2 m/s. The

15

corresponding value for the 87 µm sized Martian grains can be found on fig-ure 4 and reads uÄth ∼ 0.64 m/s. The Mars to Earth ratios for the fluid andgrain densities, as well as the gravity acceleration are known. As discussed inthe calculation of the transport thresholds (see Appendix) one expects thatαÄ is of order unity and for simplicity we simply take αÄ = αÊ. Finally, themost speculative part naturally concerns the value of ηÄ. Assuming that thewinds on Mars are similar to those on Earth, we computed on our wind datafrom Atlantic Sahara the fraction of time during which u∗ > 0.64 m/s andgot ∼ 3%. This value corresponds to few days per year and is probably re-alistic as, besides, the soil of Mars is frozen during the winter season. Withthese numerical values, we find that σÄ is smaller than σÊ by more than fivedecades. In other words, the typical time over which we could see a significantevolution of the bedforms on martian dunes is of the order of tens to hundredscenturies. Similarly, the ratio cÄ/cÊ is of the order of 10

−4. Note that theseMars to Earth time and velocity ratios are proportional to ηÄ, so that larger orsmaller values for this speculative parameter do not change dramatically theconclusions, the main contribution being the density ratio to the power 2 or3 (see equations (9–10)). Therefore, as some satellite high resolution picturesdefinitively show some evidence of aeolian activity – e.g. avalanche outlines –it may well be that the martian dunes are fully active but not significantly atthe the human scale.

5 Conclusion

As for the last section, we would like to conclude the paper with a summary ofthe status of the numerous hypothesis and facts we have mixed and discussed.Although coming from recent theoretical works, it is now well accepted thatthe wavelength λ at which dunes form from a flat sand bed is governed bythe so-called saturation length Lsat. However, the dependences of Lsat withthe numerous control parameters (Shields number, grain and flow Reynoldsnumbers, Galileo number, grain to fluid density ratio, finite size effects, etc)is still a matter of debate. In this paper, we have collected measurements ofλ in various situations that where previously thought of as disconnected: sub-aqueous ripples, micro-dunes in Venus wind tunnel, fresh snow dunes, aeoliandunes on Earth and dunes on Mars. We show that the averaged wavelength(and thus Lsat) is proportional to the grain size times the grain to fluid den-sity ratio. This does not preclude sub-dominant dependencies of Lsat with theother dimensionless parameters. In each situation listed above, we have facedspecific difficulties.Subaqueous ripples — The transport mechanism under water, namely the di-rect entrainment by the fluid, is different from the four other situations. Thesaturation process could thus be very different in this case. The longest re-

16

laxation length could for instance be controlled by the grain sedimentation,as recently suggested by Charru [25]. As there is also negative feedback oftransport on the flow, the destabilization wavelength would be larger thanour prediction. The formation of subaqueous ripples has remained controver-sial mostly because the experimental data are very dispersed. Most of themsuffer from finite size effects (water depth or friction on lateral walls), from un-controlled entrance conditions leading to an inhomogeneous destabilization, orfrom a late determination of the wavelength after a pattern coarsening period.For the furthest left data point only, we are certain that all these problemswere avoided.‘Venus’ micro-dunes — This beautiful experiment gives the most disperseddata of the plot. Part of it may be due to the defects listed just above. The com-plex variation of the wavelength with the wind speed observed experimentallycould be interpreted as a sub-dominant dependance of the saturation lengthon the Shields number. However, not only the size of the dunes changes butalso their shapes. This is definitely the signature of a second length scale atwork.Fresh snow dunes — On the basis of existing photographs, we have clearlyidentified snow dunes pattern resembling sand aeolian destabilization ones.They form under strong wind, on icy substrate. The obvious difficulty is thatthe precise state of the snow flakes during the dune formation involves com-plex thermodynamical processes that do not exist in the other cases.Aeolian dunes on Earth — Although the instability takes place in the fieldunder varying wind conditions, the resulting wavelength is very robust fromplace to place and time to time. We consider this point as the reference onein the plot.Martian dunes — The specific difficulty of the Martian case does not comefrom the wavelength measurements, as the available photographs are well re-solved in space, but rather from the prior determination of the diameter d ofthe grains involved. The controversy comes from early determinations basedon thermal diffusivity measurements, concluding that dunes are covered bylarge (500 µm) grains. Such large grains would lead to a significant shift ofthe data point towards the right of figure 5. A large part of this paper isconsequently devoted to an independent determination of d, based on theanalysis of the Martian rovers photographs. Our determination is very differ-ent: 87 ± 25 µm. We have computed the sand transport phase diagram withmore subtleties than in the available literature (including in particular hys-teresis and cohesion) and shown that this size corresponds to saltating grains.A directly related controversial point was the state of the Martian dunes (stillactive or fossile). We have shown that moderately large winds can transportthe grains (the dynamical threshold is much lower than the static one) but thatthe characteristic time scale over which dunes form is five orders of magnitudelarger than on Earth.

17

The Moroccan wavelength histogram has been obtained in collaboration withHicham Elbelrhiti. We thank Éric Clément and Évelyne Kolb for useful ad-vices on the way cohesion forces can be estimated. We thank François Charruand Douglas Jerolmack for discussions. We thank Brad Murray for a carefulreading of the manuscript. Part of this work is based on lectures given duringthe granular session of the Institut Henri Poincaré in 2005. This study wasfinancially supported by an ‘ACI Jeunes Chercheurs’ of the french ministry ofresearch.

18

A A model for transport thresholds, including cohesion

We have directly measured the grain diameter (87±25 µm) in Martian ripplesphotographed by the rovers. We then made the double assumption that (i)the grains composing the martian dunes are of the same size and that (ii)these grains participate to saltation transport whenever the wind is sufficientlystrong. In order to support hypothesis (ii), we have computed the transportthresholds in the Martian conditions. We find that the grains which move firstfor an increasing wind are around 65 µm in diameter. The full discussion of thetransport thresholds is however too heavy to be incorporated into the bodyof the article, mainly devoted to the dune wavelength scaling law. We givebelow a short but self-sufficient derivation of the scaling laws for the staticand dynamic transport thresholds.

A.1 Definition of transport thresholds, hysteresis

We consider the generic case of a fluid boundary layer over a flat bed composedof identical sand grains. For given grains and surrounding fluid, the shear stressτ controls the sand transport.The dominant mechanism for grain erosion de-pends on the sand to fluid density ratio. In dense fluids grains are directlyentrained by the flow whereas in low density fluids grains are mostly splashedup by other grains impacting the sand bed. Two thresholds are associated tothese two mechanisms: starting from a purely static sand bed, the first grainis dragged from the bed and brought into motion at the static threshold τsta.Once sand transport is established, it can sustain by the collision/ejectionprocesses down to a second threshold τdyn. The sand transport thus presentsan hysteresis – responsible for instance for the formation of streamers. Fi-nally, when the fluid velocity becomes sufficiently high, more and more grainsremain in suspension in the flow, trapped in large velocity fluctuations. Al-though there is no precise threshold associated to suspension, one expect thatthese fluctuations becomes dominant for the grain trajectories when the shear

velocity u∗ =√

τ/ρf is much larger than the grain sedimentation velocity ufall.The behaviour of these thresholds in the Martian atmosphere, as well as inwater and air, with respect to the grain diameter is summarized in figures 4and A.2, and we shall now discuss how to compute these curves.

A.2 An analytic expression for the turbulent boundary layer velocity profile

In the dune context, the flow is generically turbulent far from the sand bedand the velocity profile is known to be logarithmic. However, our problem is

19

more complicated as we have to relate the velocity u of the flow around thesand grains (i.e. the velocity at the altitude, say, z = d/2 that we call v in thefollowing) to the shear velocity u∗. Depending on the grain based Reynoldsnumber, there either exists a viscous sub-layer between the soil and the fullyturbulent zone, or the momentum transfer is directly due to the fluctuationsinduced by the soil roughness. The first step is thus to derive an expressionfor the turbulent boundary layer wind profile valid in the two regimes, far orclose to the bed. To do so, we express the shear stress as the sum of a visousand a turbulent part as

τ = ρfν∂zu+ κ2(z + rd)2ρf |∂zu|∂zu, (A.1)

where ν is the kinematic viscosity of the fluid, κ the von Kármán constantand r the aerodynamic roughness rescaled by the grain diameter d. The shearstress is constant all through the turbulent boundary layer and equal to ρfu

2∗.

Let us define the grain Reynolds number as:

Re0 =2κu∗rd

ν. (A.2)

and a non-dimensional distance to the bed:

Z = 1 +z

rd(A.3)

Note that Z is tends to 1 on the sand bed and that it is equal to 1 + 1/2rat the center of the grain. With this notations, the differential equation (A.1)can be rewritten under the form:

Z2(

d(κu/u∗)

dZ

)2

+2

Re0

(

d(κu/u∗)

dZ

)

− 1 = 0, (A.4)

which is easily integrated into:

u =u∗κ

sinh−1(Re0Z̄) +1−

√

1 +Re20Z̄2

Re0Z̄

Z̄=Z

Z̄=1

. (A.5)

Performing expansions in the purely viscous and turbulent regimes, one canshow that a good approximation of the relation between the shear velocity u∗and typical velocity around the grain v is given by

u2∗ =2ν

dv +

κ2

ln2(1 + 1/2r)v2. (A.6)

20

Fig. A.1. a Grain packing geometry considered for the computation of the statictransport thresholds. b Schematic drawing showing the contact between grains atthe micron scale.

A.3 Static threshold: influence of Reynolds number

The bed load (tractation) threshold is directly related to the fact that thegrains are trapped in the potential holes created by the grains at the sand bedsurface. To get the scaling laws, the simplest geometry to consider is a singlespherical grain jammed between the two neighbouring (fixed) grains below it,see figure A.1a. Let us first discuss the situation in which the cohesive forcesbetween the grains are negligible and the friction at the contacts is sufficient toprevent sliding. It can be inferred from figure A.1a that the loss of equilibriumoccurs for a value of the driving force F proportional to the submerged weightof the grain: F ∝ (ρs− ρf )gd

3, where ρs is the mass density of the sand grain,and ρf is that of the fluid – which is negligible with respect to ρs in the caseof aeolian transport. As F is proportional to the shear force τd2 exerted bythe fluid, the non-dimensional parameter controlling the onset of motion isthe Shields number, which characterises the ratio of the driving shear stressto the normal stress:

Θ =τ

(ρs − ρf )gd. (A.7)

The threshold value can be estimated from the geometry of the piling, anddepends on whether rolling or lifting is the mechanism which makes the grainmove. Finally, the local slope of the bed modifies the threshold value as trapsbetween the grains are less deep when the bed is inclined. In particular, itsvalue must vanish as the bed slope approaches the (tangent of the) avalancheangle. We here ignore these refinements which can be incorporated into thevalues of A and B (see section 1).

At the threshold, the horizontal force balance on a grain of the bed reads

π

6µ(ρs − ρf )gd

3 =π

8Cdρfv

2thd

2, (A.8)

where µ is a friction coefficient, and Cd the drag coefficient which is a functionof the grain Reynolds number. With a good accuracy, the drag law for natural

21

grains can be put under the form:

Cd =(

C1/2∞ + s

√

ν

vd

)2

(A.9)

with C∞ ≃ 1 and s ≃ 5 for natural sand grains [44].

At this stage, we introduce the viscous size dν, defined as the diameter ofgrains whose free fall Reynolds number ufalld/ν is unity:

dν = (ρs/ρf − 1)−1/3 ν2/3 g−1/3 (A.10)

It corresponds to a grain size at which viscous and gravity effects are of thesame order of magnitude. We define ṽ as the fluid velocity at the scale of thegrain normalised by (ρs/ρf − 1)

1/2(gd)1/2. From the three previous relations,we get the equation for ṽth:

C1/2∞ ṽth + s

(

dνd

)3/4

ṽ1/2th −

(

4µ

3

)1/2

= 0, (A.11)

which solves into

ṽth =1

4C∞

s2(

dνd

)3/2

+ 8(

µC∞3

)1/2

1/2

− s

(

dνd

)3/4

2

. (A.12)

The expression of the static threshold Shields number is finally:

Θ∞th = 2

(

dνd

)3/2

ṽth +κ2

ln2(1 + 1/2r)ṽ2th. (A.13)

A.4 Static threshold: influence of cohesion

For small grains, the cohesion of the material strongly increases the staticthreshold shear stress. Evaluating the adhesion force between grains is a dif-ficult problem in itself. We consider here two grains at the limit of separationand we assume that the multi-contact surface between the grains has beencreated with a maximum normal load Nmax, see figure A.1b. The adhesionforce Nadh can be expressed as an effective surface tension γ̃ times the radiusof curvature of the contact, which is assumed to scale on the grain diameter:

Nadh ∝ γ̃d. (A.14)

22

This effective surface tension is much smaller than the actual one, γ, as thereal area of contact Areal is much smaller than the apparent one AHertz.:

γ̃ = γArealAHertz

. (A.15)

The apparent area of contact can be computed following Hertz law for twospheres in contact under a load Nmax:

AHertz ∝

(

Nmaxd

E

)2/3

, (A.16)

where E is the Young modulus of the grain [45]. To express the real area ofcontact, we need to know whether the micro-contacts are in an elastic or aplastic regime. Within a good approximation, Areal can expressed in both cases[46] as:

Areal ∝NmaxM

, (A.17)

where M is the Young modulus E (elastic regime) or the hardness H (plasticregime) of the material. Altogether, we then get:

Nadh ∝ γE2/3

M(Nmaxd)

1/3. (A.18)

In order to bring into motion such grains, the shear must be large enough toovercome both weight and adhesion, so that, for Nmax ∼ ρsgd

3 (i.e. the weightof one grain) the critical Shields number is the sum of two terms and takesthe form of

Θth = Θ∞th

1 +3

2

(

dmd

)5/3

, (A.19)

with

dm ∝(

γ

M

)3/5(

E

ρsg

)2/5

. (A.20)

Note that, by contrast to the references [31,32,33,34,35], we find that theadhesive force finally scales, for d → 0 with d4/3 and the critical Shieldsnumber with d−5/3 (instead of exponents 1 and −2 respectively).

A.5 Dynamical threshold

The dynamical threshold can be defined as the value of the control parameterfor which the saturated flux vanishes. Fitting the flux vs shear velocity relation,one can measure the dynamical threshold in a very precise way (see for instancethe analysis in [20] of the data obtained by Rasmussen et al. [22]).

23

Fig. A.2. Diagram showing the mode of transport in the aeolian (a) and underwater(b) cases, as a function of the grain diameter d and of the turbulent shear velocityuth (left) or of the wind speed Uth at 2 m above the soil (right). The dynamicalthreshold (dashed line) is below the static threshold (solid line) in the aeolian casebut much above it underwater. The dark gray is the zone where no transport ispossible. Above, the background color codes for the ratio u∗/ufall: white correspondsto negligible fluctuations and gray to suspension. The experimental points are takenfrom (◦) Chepil [48] and (�) Rasmussen [22,20] in the aeolian case and from (◦)Yalin and Karahan [49] in the underwater case.

As shown in [20], the wind velocity profile is almost undisturbed at the dynam-ical threshold. The shear stress threshold is thus achieved when the velocitythat a grain acquires after a single jump is just sufficient to eject on averageone single grain out of the bed after a collision (unit replacement capacity cri-terion). The impact velocity at this dynamical threshold is thus proportionalto the trapping velocity i.e. the velocity needed for a grain to escape from itspotential trapping (see Quartier et al. [47]):

v↓ = a

√

√

√

√

√gd

1 +3

2

(

dmd

)5/3

, (A.21)

An analytical expression of the dynamical threshold has been derived in [20],but only in the limit of large Reynolds numbers. To generate the plots dis-played in figures 4 and A.2, we use here a numerical integration of the equa-tions of motion of a grain – with equations (A.5) and (A.9) for the wind profileand drag coefficient. The trajectories are computed for an ejection velocity v↑equal to e v↓ and a typical ejection angle π/4 (see [20]) .

A.6 From Earth to Mars

In order to get a reasonable transport diagram on Mars, we have first tuned thedifferent parameters controlling the dynamic and static thresholds to repro-

24

duce within the same model the underwater and aeolian data (figure A.2). Forthe friction coefficient µ, we have simply taken the avalanche slope for typicalaeolian grains tan(32◦). As already found in [20], the rescaled soil roughness ris higher than the value found by Bagnold. It is here adjusted to r = 1/10. Thevalue of a, the impact velocity needed to eject statistically one grain rescaledby the trapping velocity, is adjusted to 15. The restitution coefficient was ad-justed to e = 2/3. For obvious reasons it is possible to obtain almost the samecurves with different pairs (a, e). Finally the cohesion length dm was tuned to25 µm, which is consistent with the above calculation.

We see in figure A.2 that the agreement is excellent both in the aeolian andunderwater cases. Due to the low density ratio in water, the collision processis completely inefficient. As a consequence, the dynamical threshold is wellabove the static one: the only erosion mechanism is the direct entrainmentof grains by the fluid (tractation). On the other hand, the static threshold– computed with the same formula as that obtained in water – is well abovethe experimental data in the aeolian case: there is a very important hysteresisin that case. With both sets of data, we have thus a complete calibration ofthe dynamic and static threshold parameters, which allows to a good degreeof confidence for the computation of the transport diagram on Mars (figure 4).

25

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27

Dune instabilitySize of saltating grains on MarsDirect measurement of grain sizes

A dune wavelength scaling lawDiscussion: time scales and velocity for the martian dunesConclusionA model for transport thresholds, including cohesionDefinition of transport thresholds, hysteresisAn analytic expression for the turbulent boundary layer velocity profileStatic threshold: influence of Reynolds numberStatic threshold: influence of cohesionDynamical thresholdFrom Earth to Mars

References